Introduction

Our primary goal here is to develop and test via simulation a bank of CDI items and IRT parameters that we can recommend to those wanting to develop and conduct CDI computerized adaptive tests (CATs). Our approach is as follows: We first fit basic IRT models (1-parameter logistic (1PL; i.e. Rasch), 2PL, and 3PL) to CDI data (French (French), WS form), and perform a model comparison. For the favored model, we then identify candidate items for removal based on low total item information, and then use the (full) item bank in a variety of computerized adaptive test (CAT) simulations on wordbank data. We provide recommendations for CAT algorithms and stopping rules to be passed on to CAT developers, and benchmark CAT performance compared to random baselines tests of a similar length.

Data

We use the combined French (French) production data from Words & Gestures (WG, who are already producing words) and Words & Sentences (WS) CDI form, for a total of 1782 participants. From this data we remove 121 children <12 months of age, who should not be producing any words yet. We also remove an additional 56 children 12+ months of age who are not yet producing any words, as these children cannot be used to fit the IRT models. The production sumscores by age for the remaining children are shown below.

Age N
12 35
13 91
14 39
15 26
16 85
17 57
18 37
19 61
20 68
21 95
22 104
23 114
24 168
25 92
26 58
27 65
28 60
29 76
30 67
31 29
32 49
33 42
34 42
35 33
36 10
37 1
39 1

IRT Models

We fit each type of basic IRT model (Rasch, 2PL, and 3PL) using the mirt package.

Model comparison.

Compared to the Rasch model, the 2PL model fits better and is preferred by both AIC and BIC.

Comparison of Rasch and 2PL models.
Model AIC BIC logLik df
Rasch 404496.6 408230.9 -201554.3 NaN
2PL 391291.6 398749.5 -194259.8 692

The 2PL is favored over the 3PL model by both AIC and BIC.

Comparison of 2PL and 3PL models.
Model AIC BIC logLik df
2PL 391291.6 398749.5 -194259.8 NaN
3PL 392257.9 403444.8 -194050.0 693

The 2PL is preferred over both the Rasch (1PL) model and the 3PL model, so we do the rest of our analyses using the 2PL model as the basis for the CAT. Next we look for linear dependencies (LD) among the items, and also check for ill-fitting items. We will remove any items that show both strong LD and poor fit.

Item bank

Examine Linear Dependencies

We examined each item for pairwise linear dependencies (LD) with other items using \(\chi^{2}\) (Chen & Thissen, 1997), and found that 682 items show strong LD (Cramer’s \(V \geq 0.5\)). This suggests multidimensionality, meaning we may want to look into exploratory factor analysis.

Ill-fitting items

Our next goal is to determine if all items should be included in the item bank. Items that have very bad properties should probably be dropped. We will prune any ill-fitting items (\(\chi^{2}\) \(p<.001\)) from the full 2PL model that also showed strong LD.

28 items did not fit well in the full 2PL model, and these items are shown below.

Now we re-fit the 2PL model without the 28 items showing strong LD and poor fit removed. (Note: should look removed/remaining items by category, difficulty, discrimination, and RMSEA.)

Plot Item Parameters

Next, we examine the coefficients of the 2PL model. Items that are estimated to be very easy (e.g., maman, papan, au revoir) or very difficult (au sujet de, au sommet de) are highlighted, as well as those at the extremes of discrimination (a1).

## Warning: ggrepel: 163 unlabeled data points (too many overlaps). Consider
## increasing max.overlaps

Next, we will run simulated CATs on the data from the 1605 12-36 month-olds. However, since many of these participants’ data are from the CDI:WG form, there are many missing responses (compared to the CDI:WS). In order to run the simulated CATs, we impute the missing data using the participants’ estimated ability and the 2PL model. Overall, 27.4% of the data was missing, and will be imputed.

CAT Simulations

For each wordbank subject, we simulate a CAT using a maximum of 25, 50, 100, 200, 300, or 400 items, with the termination criterion that it reach an estimated SEM of .1. For each of these simulations, we examine 1) which items were never used, 2) the median and mean number of items used, 3) the correlation of ability scores estimated from the CAT and from the full CDI, and 4) the mean standard error of the CATs.

CAT simulations with 2PL model compared to full CDI.
Maximum Qs Median Qs Asked Mean Qs Asked r with full CDI Mean SE Reliability Items Never Used
25 25 22.646 0.992 0.159 0.975 485
50 43 35.832 0.994 0.143 0.980 427
75 43 47.137 0.995 0.137 0.981 384
100 43 57.687 0.995 0.135 0.982 337
200 43 97.330 0.995 0.132 0.983 193
300 43 135.178 0.996 0.130 0.983 84
400 43 172.206 0.996 0.130 0.983 16

Finally, following Makransky et al. (2016), we run a series of fixed-length CAT simulations and again compare the thetas from these CATs to the ability estimates from the full CDI. The results are quite good even for 25- and 50-item tests, but note that we add a comparison to tests of randomly-selected questions (per subject), and find that ability estimates from these tests are also strongly correlated with thetas from the full CDI. The mean standard error of the random tests shows more of a difference.

Fixed-length CAT simulations with 2PL model compared to full CDI.
Test Length r with full CDI Mean SE Reliability Items Never Used Random Test r with full CDI Random Test Mean SE
25 0.992 0.155 0.976 464 0.974 0.307
50 0.995 0.127 0.984 308 0.983 0.249
75 0.996 0.117 0.986 223 0.987 0.219
100 0.996 0.111 0.988 136 0.990 0.200
200 0.997 0.102 0.990 0 0.993 0.157
300 0.997 0.098 0.990 0 0.995 0.133
400 0.997 0.097 0.991 0 0.996 0.118

Preferred CAT Settings

Testing with a minimum of 25 items, a maximum of 50, and termination at SE = .1, and ML scoring. First we’ll do it using the MI start item, and then we’ll try choosing an age-based starting item per subject (based on mean theta for each age).

We select a starting item with a difficulty just below the average ability (theta) for each age (in months). The mean theta per age is shown below, along with the selected starting item.

age theta sd n definition index item_info
12 -2.49 0.67 35 au.revoir 14 1.11
13 -2.50 0.57 91 au.revoir 14 1.09
14 -2.38 0.64 39 au.revoir 14 1.27
15 -1.71 0.57 26 eau..drink. 71 3.12
16 -1.43 1.40 85 couche 151 6.04
17 -1.25 1.58 57 couche 151 8.22
18 -2.17 0.51 37 bravo 18 1.53
19 -1.49 1.34 61 couche 151 5.07
20 -0.20 1.34 68 parler 654 9.43
21 -0.09 1.35 95 se.réveiller 659 6.94
22 -0.44 1.30 104 parler 654 8.81
23 -0.53 1.28 114 cuisine 463 8.40
24 0.11 1.24 168 construire 616 6.37
25 -0.04 1.31 92 renverser 639 6.80
26 0.16 1.28 58 se.dépêcher 349 6.18
27 0.04 1.16 65 renverser 639 6.76
28 0.69 1.02 60 souhaiter 650 6.57
29 0.42 1.13 76 penser 657 5.57
30 0.40 1.20 67 long.ue 601 5.60
31 -0.28 0.50 29 parler 654 10.63
32 -0.21 0.63 49 parler 654 9.80
33 -0.16 0.50 42 parler 654 8.55
34 -0.01 0.48 42 renverser 639 6.86
35 -0.01 0.57 33 renverser 639 6.86
36 -0.04 0.73 10 renverser 639 6.81
37 -0.06 NA 1 renverser 639 6.74
39 0.10 NA 1 construire 616 6.43
CAT simulations with min=25, max=50, stopping at SE=0.15.
Scoring / Start Item Median Qs Asked Mean Qs Asked r with full CDI Mean SE Reliability Items Never Used
ML / MI 25 34.125 0.993 0.158 0.975 417
MAP / MI 25 33.652 0.994 0.143 0.980 422
ML / age-based 25 33.946 0.993 0.157 0.975 405
MAP / age-based 25 33.525 0.994 0.142 0.980 409

Age analysis

Does the CAT show systematic errors with children of different ages? The table below shows correlations between ability estimates from the full CDI compared to the estimated ability from each fixed-length CAT split by age (139 11-13 month-olds, 155 14-16 mos, 147 17-19 mos, 276 20-22 mos, 370 23-25 mos, 192 26-28 mos, 173 29-31 mos, 124 32-35 mos, and 28 35-38 mos). This is comparable to Table 3 of Makransky et al. (2016), and the correlations here are consistently high across age groups.

Correlation between fixed-length CAT ability estimates and the full CDI.
Test Length [11,14) mos [14,17) mos [17,20) mos [20,23) mos [23,26) mos [26,29) mos [29,32) mos [32,35) mos [35,38] mos
25 0.927 0.990 0.993 0.994 0.992 0.986 0.984 0.963 0.988
50 0.961 0.995 0.997 0.996 0.994 0.991 0.990 0.973 0.990
75 0.969 0.996 0.997 0.996 0.994 0.992 0.991 0.985 0.994
100 0.975 0.997 0.998 0.997 0.995 0.992 0.993 0.989 0.995
200 0.985 0.998 0.999 0.998 0.996 0.993 0.994 0.997 0.998
300 0.994 0.998 0.999 0.998 0.996 0.993 0.995 0.999 0.999
400 0.997 0.999 0.999 0.998 0.996 0.993 0.995 0.999 1.000

We further look at the correlations with age using the preferred CAT settings (min_items=25, max_items=50, stopping at SE=.15).

Correlation between the preferred CAT’s ability estimates and the full CDI.
Scoring / Start Item [11,14) mos [14,17) mos [17,20) mos [20,23) mos [23,26) mos [26,29) mos [29,32) mos [32,35) mos [35,38] mos
ML / MI 0.955 0.991 0.993 0.995 0.992 0.988 0.987 0.962 0.986
MAP / MI 0.953 0.993 0.995 0.995 0.993 0.989 0.987 0.962 0.988
ML / age-based 0.949 0.992 0.993 0.995 0.992 0.988 0.987 0.966 0.986
MAP / age-based 0.955 0.994 0.995 0.995 0.993 0.99 0.988 0.965 0.988

Below we show the distribution of ability (theta) from the 2PL model by age.

Ability analysis

Finally, we ask whether the fixed-length CATs work well for children of different abilities. Below are scatterplots that show the standard error estimates vs. estimated ability (theta) for each child on the different simulated fixed-length CATs. The 25-item CAT shows some visible distortion, but the 50-item CAT is already quite smooth, and the 75-item CAT indistinguishable from the 300- or 400-item CATs. Based on these plots and the above tables we may recommend that users adopt a 50-item CAT using the 2PL parameters, but suggest that they may want to administer a full CDI if the participant’s estimated theta from the CAT is <-0.5 or >2 (where the SE from CAT starts to exceed 0.1).

Item selection for item bank

Of the 665 pruned CDI:WS items, 357 were selected on one or more administrations of the fixed-length 50-item CATs simulated from the wordbank data. Which items were most frequently selected for the fixed-length 50-item CAT? Shown in the table below, only 3 items were selected on more at least 50% of the tests.

Items chosen on at least 50% of the 50-item CATs.
Item Proportion
penser 1.00
renverser 0.61
parler 0.55

Below we show the overall distribution of how many of the 665 pruned CDI:WS items were selected on what percent of the CATs of varying length (50, 75, or 100 items). Note that we do not include in the graph the number of items that were never selected on each test: 308 items never selected on the 50-item test, 223 items on the 75-item test, and 136 items never selected on the 100-item test. The longer the test, the less skewed the distribution, but even on the 100-item CAT most of the appearing items are selected less than a third of the time.

Below we show the 84 items from the pruned CDI:WS that were never selected on the maximum 300-item CAT.

petit.déjeuner frigo jeter et
animal lavabo mettre rue.route
glace salon nager copain.ine
œuf balançoire nettoyer peux
petits.pois ciel tirer faire
raisin école toucher laisse.moi
soupe jardin voir panier
sucre magasin derrière coussins
figure pelle sur noir
langue piscine ma.mon.mes fort.e
culotte.slip travail tous.tout orange
robe content.e un.peu triste
short gentil.le jeu blanc.he
tee.shirt joli haricot attraper
bouteille méchant.e haricots.verts aimer.bien
couverture propre machine.à.laver faire.de.la.peinture
médicaments sec.che les.courses glisser
papier aider à.côté.de travailler
serviette chanter un.une goûter..verb.
fauteuil essuyer Il cuisiner
four faire.du.vélo.de.la.moto elle aller

What about the items that are most selected across all of the CATs (25-400-item)? Here are the top 50:

penser camping chaque ceux venteux
renverser avoir.à.faire étendage garderie couche
parler yeux aucun.e pourrait devoir.faire
cuisine sombre chewing.gum esquimau hamburger
détester Centre.ville pop.corn au.sujet.de flan
assiette toi.même aller.bien.avec recevoir trotteur
tendre faire.du.patin ordures cuisinière tricycle
au.sommet.de limonade cornflakes goutter grenouillère
moi.même infirmière vers beignet donc
station.service au.loin déposer souhaiter par.dessus

These are predominantly nouns, including several body parts.

Example CAT

We now show an example CAT for two simulated participants, one with ability (theta) = 0, and one with theta = 1. The CAT gives a minimum of 25 questions and terminates either when SEM=0.15 or when 50 items is reached. The theta estimates over the test for each participant is shown below, with selected item indices on the x axis. The theta=0 participant (left) answered 27 questions, and the theta=1 participant (right) answered 50. The final estimated theta for the theta=0 participant was -0.011, and for the theta=1 participant was 0.813. The package mirtCAT can be directly used to simply generate a web interface (Shiny app) that allows such CATs to be run on real participants, as well as the simulations we have conducted here.

References

Makransky, G., Dale, P. S., Havmose, P. and Bleses, D. (2016). An Item Response Theory–Based, Computerized Adaptive Testing Version of the MacArthur–Bates Communicative Development Inventory: Words & Sentences (CDI:WS). Journal of Speech, Language, and Hearing Research. 59(2), pp. 281-289.